Integrand size = 22, antiderivative size = 114 \[ \int \frac {1+x}{(-1+x) \sqrt [3]{-x^2+x^3}} \, dx=-\frac {6 \left (-x^2+x^3\right )^{2/3}}{(-1+x) x}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x^2+x^3}}\right )-\log \left (-x+\sqrt [3]{-x^2+x^3}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{-x^2+x^3}+\left (-x^2+x^3\right )^{2/3}\right ) \] Output:
-6*(x^3-x^2)^(2/3)/(-1+x)/x+3^(1/2)*arctan(3^(1/2)*x/(x+2*(x^3-x^2)^(1/3)) )-ln(-x+(x^3-x^2)^(1/3))+1/2*ln(x^2+x*(x^3-x^2)^(1/3)+(x^3-x^2)^(2/3))
Time = 0.17 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.15 \[ \int \frac {1+x}{(-1+x) \sqrt [3]{-x^2+x^3}} \, dx=\frac {x^{2/3} \left (-12 \sqrt [3]{x}+2 \sqrt {3} \sqrt [3]{-1+x} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{2 \sqrt [3]{-1+x}+\sqrt [3]{x}}\right )-2 \sqrt [3]{-1+x} \log \left (\sqrt [3]{-1+x}-\sqrt [3]{x}\right )+\sqrt [3]{-1+x} \log \left ((-1+x)^{2/3}+\sqrt [3]{-1+x} \sqrt [3]{x}+x^{2/3}\right )\right )}{2 \sqrt [3]{(-1+x) x^2}} \] Input:
Integrate[(1 + x)/((-1 + x)*(-x^2 + x^3)^(1/3)),x]
Output:
(x^(2/3)*(-12*x^(1/3) + 2*Sqrt[3]*(-1 + x)^(1/3)*ArcTan[(Sqrt[3]*x^(1/3))/ (2*(-1 + x)^(1/3) + x^(1/3))] - 2*(-1 + x)^(1/3)*Log[(-1 + x)^(1/3) - x^(1 /3)] + (-1 + x)^(1/3)*Log[(-1 + x)^(2/3) + (-1 + x)^(1/3)*x^(1/3) + x^(2/3 )]))/(2*((-1 + x)*x^2)^(1/3))
Time = 0.24 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.88, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2467, 87, 71}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x+1}{(x-1) \sqrt [3]{x^3-x^2}} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt [3]{x-1} x^{2/3} \int \frac {x+1}{(x-1)^{4/3} x^{2/3}}dx}{\sqrt [3]{x^3-x^2}}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {\sqrt [3]{x-1} x^{2/3} \left (\int \frac {1}{\sqrt [3]{x-1} x^{2/3}}dx-\frac {6 \sqrt [3]{x}}{\sqrt [3]{x-1}}\right )}{\sqrt [3]{x^3-x^2}}\) |
\(\Big \downarrow \) 71 |
\(\displaystyle \frac {\sqrt [3]{x-1} x^{2/3} \left (-\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x-1}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )-\frac {6 \sqrt [3]{x}}{\sqrt [3]{x-1}}-\frac {3}{2} \log \left (\frac {\sqrt [3]{x-1}}{\sqrt [3]{x}}-1\right )-\frac {\log (x)}{2}\right )}{\sqrt [3]{x^3-x^2}}\) |
Input:
Int[(1 + x)/((-1 + x)*(-x^2 + x^3)^(1/3)),x]
Output:
((-1 + x)^(1/3)*x^(2/3)*((-6*x^(1/3))/(-1 + x)^(1/3) - Sqrt[3]*ArcTan[1/Sq rt[3] + (2*(-1 + x)^(1/3))/(Sqrt[3]*x^(1/3))] - (3*Log[-1 + (-1 + x)^(1/3) /x^(1/3)])/2 - Log[x]/2))/(-x^2 + x^3)^(1/3)
Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[d/b, 3]}, Simp[(-Sqrt[3])*(q/d)*ArcTan[2*q*((a + b*x)^(1/3)/( Sqrt[3]*(c + d*x)^(1/3))) + 1/Sqrt[3]], x] + (-Simp[3*(q/(2*d))*Log[q*((a + b*x)^(1/3)/(c + d*x)^(1/3)) - 1], x] - Simp[(q/(2*d))*Log[c + d*x], x])] / ; FreeQ[{a, b, c, d}, x] && PosQ[d/b]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.48 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.35
method | result | size |
risch | \(-\frac {6 x}{\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}+\frac {3 \left (-\operatorname {signum}\left (-1+x \right )\right )^{\frac {1}{3}} x^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x\right )}{\operatorname {signum}\left (-1+x \right )^{\frac {1}{3}}}\) | \(40\) |
meijerg | \(-\frac {3 \left (-\operatorname {signum}\left (-1+x \right )\right )^{\frac {1}{3}} x^{\frac {1}{3}}}{\operatorname {signum}\left (-1+x \right )^{\frac {1}{3}} \left (1-x \right )^{\frac {1}{3}}}-\frac {3 \left (-\operatorname {signum}\left (-1+x \right )\right )^{\frac {1}{3}} x^{\frac {4}{3}} \operatorname {hypergeom}\left (\left [\frac {4}{3}, \frac {4}{3}\right ], \left [\frac {7}{3}\right ], x\right )}{4 \operatorname {signum}\left (-1+x \right )^{\frac {1}{3}}}\) | \(54\) |
pseudoelliptic | \(-\frac {\left (\sqrt {3}\, \arctan \left (\frac {\left (2 \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )+\ln \left (\frac {\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}-x}{x}\right )-\frac {\ln \left (\frac {\left (\left (-1+x \right ) x^{2}\right )^{\frac {2}{3}}+\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )}{2}\right ) \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}+6 x}{\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}\) | \(102\) |
trager | \(-\frac {6 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}}{\left (-1+x \right ) x}-3 \ln \left (\frac {45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}-90 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x -72 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}-72 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x -87 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}+69 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x +15 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}+15 x \left (x^{3}-x^{2}\right )^{\frac {1}{3}}+20 x^{2}-12 x}{x}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (\frac {45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}-90 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x +72 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}+72 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x +57 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -9 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}-9 x \left (x^{3}-x^{2}\right )^{\frac {1}{3}}-4 x^{2}+x}{x}\right )+\ln \left (\frac {45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}-90 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x -72 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}-72 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x -87 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}+69 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x +15 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}+15 x \left (x^{3}-x^{2}\right )^{\frac {1}{3}}+20 x^{2}-12 x}{x}\right )\) | \(509\) |
Input:
int((1+x)/(-1+x)/(x^3-x^2)^(1/3),x,method=_RETURNVERBOSE)
Output:
-6*x/((-1+x)*x^2)^(1/3)+3/signum(-1+x)^(1/3)*(-signum(-1+x))^(1/3)*x^(1/3) *hypergeom([1/3,1/3],[4/3],x)
Time = 0.07 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.20 \[ \int \frac {1+x}{(-1+x) \sqrt [3]{-x^2+x^3}} \, dx=-\frac {2 \, \sqrt {3} {\left (x^{2} - x\right )} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{3 \, x}\right ) + 2 \, {\left (x^{2} - x\right )} \log \left (-\frac {x - {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - {\left (x^{2} - x\right )} \log \left (\frac {x^{2} + {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x + {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) + 12 \, {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{2 \, {\left (x^{2} - x\right )}} \] Input:
integrate((1+x)/(-1+x)/(x^3-x^2)^(1/3),x, algorithm="fricas")
Output:
-1/2*(2*sqrt(3)*(x^2 - x)*arctan(1/3*(sqrt(3)*x + 2*sqrt(3)*(x^3 - x^2)^(1 /3))/x) + 2*(x^2 - x)*log(-(x - (x^3 - x^2)^(1/3))/x) - (x^2 - x)*log((x^2 + (x^3 - x^2)^(1/3)*x + (x^3 - x^2)^(2/3))/x^2) + 12*(x^3 - x^2)^(2/3))/( x^2 - x)
\[ \int \frac {1+x}{(-1+x) \sqrt [3]{-x^2+x^3}} \, dx=\int \frac {x + 1}{\sqrt [3]{x^{2} \left (x - 1\right )} \left (x - 1\right )}\, dx \] Input:
integrate((1+x)/(-1+x)/(x**3-x**2)**(1/3),x)
Output:
Integral((x + 1)/((x**2*(x - 1))**(1/3)*(x - 1)), x)
\[ \int \frac {1+x}{(-1+x) \sqrt [3]{-x^2+x^3}} \, dx=\int { \frac {x + 1}{{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} {\left (x - 1\right )}} \,d x } \] Input:
integrate((1+x)/(-1+x)/(x^3-x^2)^(1/3),x, algorithm="maxima")
Output:
integrate((x + 1)/((x^3 - x^2)^(1/3)*(x - 1)), x)
Time = 0.24 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.65 \[ \int \frac {1+x}{(-1+x) \sqrt [3]{-x^2+x^3}} \, dx=-\sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {6}{{\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}}} + \frac {1}{2} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right ) - \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \] Input:
integrate((1+x)/(-1+x)/(x^3-x^2)^(1/3),x, algorithm="giac")
Output:
-sqrt(3)*arctan(1/3*sqrt(3)*(2*(-1/x + 1)^(1/3) + 1)) - 6/(-1/x + 1)^(1/3) + 1/2*log((-1/x + 1)^(2/3) + (-1/x + 1)^(1/3) + 1) - log(abs((-1/x + 1)^( 1/3) - 1))
Timed out. \[ \int \frac {1+x}{(-1+x) \sqrt [3]{-x^2+x^3}} \, dx=\int \frac {x+1}{{\left (x^3-x^2\right )}^{1/3}\,\left (x-1\right )} \,d x \] Input:
int((x + 1)/((x^3 - x^2)^(1/3)*(x - 1)),x)
Output:
int((x + 1)/((x^3 - x^2)^(1/3)*(x - 1)), x)
\[ \int \frac {1+x}{(-1+x) \sqrt [3]{-x^2+x^3}} \, dx=\int \frac {x}{x^{\frac {5}{3}} \left (x -1\right )^{\frac {1}{3}}-x^{\frac {2}{3}} \left (x -1\right )^{\frac {1}{3}}}d x +\int \frac {1}{x^{\frac {5}{3}} \left (x -1\right )^{\frac {1}{3}}-x^{\frac {2}{3}} \left (x -1\right )^{\frac {1}{3}}}d x \] Input:
int((1+x)/(-1+x)/(x^3-x^2)^(1/3),x)
Output:
int(x/(x**(2/3)*(x - 1)**(1/3)*x - x**(2/3)*(x - 1)**(1/3)),x) + int(1/(x* *(2/3)*(x - 1)**(1/3)*x - x**(2/3)*(x - 1)**(1/3)),x)